Theorem reuss 3536

Description: Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.)

Assertion

Ref	Expression
reuss	|- ((A C_ B /\ E.x e. A ph /\ E!x e. B ph) -> E!x e. A ph)

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   ph(x)

Proof of Theorem reuss

Step	Hyp	Ref	Expression
1		idd 21	. . . 4 |- (x e. A -> (ph -> ph))
2	1	rgen 2679	. . 3 |- A.x e. A (ph -> ph)
3		reuss2 3535	. . 3 |- (((A C_ B /\ A.x e. A (ph -> ph)) /\ (E.x e. A ph /\ E!x e. B ph)) -> E!x e. A ph)
4	2, 3	mpanl2 662	. 2 |- ((A C_ B /\ (E.x e. A ph /\ E!x e. B ph)) -> E!x e. A ph)
5	4	3impb 1147	1 |- ((A C_ B /\ E.x e. A ph /\ E!x e. B ph) -> E!x e. A ph)

Syntax hints:   -> wi 4   /\ wa 358   /\ w3a 934   e. wcel 1710  A.wral 2614  E.wrex 2615  E!wreu 2616   C_ wss 3257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-reu 2621  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259

This theorem is referenced by: (None)